The Paris St-Germain striker glanced in Jonathan Rodriguez's cross in the 82nd minute to decide a hard-fought game.
Luis Suarez missed Uruguay's best chance of the first half, seeing a shot deflected wide with an unmarked team-mate on his shoulder.
It was the sides' first Copa America meeting since a stormy 2015 match.
Four years ago, Cavani was sent off in Uruguay's 1-0 last-eight defeat after reacting to Chile's Gonzalo Jara putting a hand on his buttocks.
Former West Brom and Nottingham Forest defender Jara was subsequently given a three-match ban.
Jara, a controversial figure in the 2015 match, kicked a pitch invader in the 2019 re-run, causing Suarez to frantically urge the referee to book the Chile defender
Chile will face Colombia, the only team to have won all three of their games in the tournament, in this year's quarter-finals after being beaten to top spot in the group by Uruguay.
Ecuador bowed out with a 1-1 draw against Japan but coach Hernan Gomez insisted his leadership was not behind his team's failure to make the last eight.
Victory against the invited Japan side would have taken Gomez's side through.
"I'm not handing over my post. Another coach will come in and the same things will happen, they are asking for results when we can't provide them," he said.
"Let's see who explodes first. I don't think I am the problem."